Over the past few decades, various types of numerical methods for the VP system have been successfully developed. The Particle-In-Cell (PIC) method employs a collection of sampled macro particles to represent the distribution function [1]}, [2]} in the Lagrangian fashion, hence avoiding the curse of dimensionality. Meanwhile, it is well-known that the PIC method suffers the inherent statistical noise. Deterministic methods are developed under the grid-based Eulerian or semi-Lagrangian (SL) framework to compute the VP system, and are becoming popular recently, see e.g. [3]}. Despite the high order accuracy for deterministic solvers, they are known to suffer from the bottleneck caused by the curse of dimensionality.
Several dimension reduction techniques have been developed.
One such example is the sparse grid approach [4]}, [5]}, [6]}, which can effectively reduce the computational complexity and is well-suited for the problems with moderately high dimensions. For the Vlasov simulations, we mention the sparse grid SL method [7]} and the sparse grid discontinuous Galerkin method [8]}, [9]}. Recently, the tensor approach emerged as a promising tool for feasible simulations of high-dimensional PDEs. Such an approach aims to extract the underlying low rank structure of the solution data with advanced tensor decompositions, potentially breaking the curse of dimensionality. The popular tensor formats include the canonical polyadic (CP) format [10]}, [11]}, [12]}, [13]}, Tucker format [14]}, [15]}, hierarchical Tucker (HT) format [16]}, [17]}, and tensor train (TT) format [18]}, [19]}, [20]}. There are several pioneering works employing the low rank tensor approach for nonlinear simulations, including the low rank SL method in the TT format [21]}, a low rank method with the CP format based on the underlying Hamiltonian formulation [22]}, a dynamical low rank method proposed in [23]}, [24]} for which the dynamical low rank approximation of the Vlasov solution is evolved on the low rank manifold using a tangent space projection, and dynamical tensor approximations for high dimensional linear and nonlinear PDEs based on functional tensor decomposition and dynamical tensor approximation [25]}.
